MIMO with MMSE equalizer

by Krishna Sankar on November 2, 2008

In a previous post, we had discussed a 2×2 MIMO transmission using BPSK modulation in Rayleigh channel with a Zero Forcing equalizer. The simulated results with the 2×2 MIMO system with zero forcing equalizer showed matching results as obtained in for a 1×1 system for BPSK modulation in Rayleigh channel. In this post, we will discuss a different equalization approach called Minimum Mean Square Error (MMSE) equalization. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

The background material on the MIMO channel has been described in the post on Zero Forcing equalizer. The text is repeated again for easy readability.

2×2 MIMO channel

In a 2×2 MIMO channel, probable usage of the available 2 transmit antennas can be as follows:

1. Consider that we have a transmission sequence, for example

2. In normal transmission, we will be sending in the first time slot, in the second time slot, and so on.

3. However, as we now have 2 transmit antennas, we may group the symbols into groups of two. In the first time slot, send and from the first and second antenna. In second time slot, send and from the first and second antenna, send and in the third time slot and so on.

4. Notice that as we are grouping two symbols and sending them in one time slot, we need only time slots to complete the transmission – data rate is doubled !

5. This forms the simple explanation of a probable MIMO transmission scheme with 2 transmit antennas and 2 receive antennas.

Figure: 2 Transmit 2 Receive (2×2) MIMO channel

Other Assumptions

1. The channel is flat fading – In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication. For a more rigorous discussion on flat fading and frequency selective fading, may I urge you to review Chapter 15.3 Signal Time-Spreading from [DIGITAL COMMUNICATIONS: SKLAR]

2. The channel experience by each transmit antenna is independent from the channel experienced by other transmit antennas.

3. For the transmit antenna to receive antenna, each transmitted symbol gets multiplied by a randomly varying complex number . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of are Gaussian distributed having mean and variance .

4. The channel experienced between each transmit to the receive antenna is independent and randomly varying in time.

5. On the receive antenna, the noise has the Gaussian probability density function with

with and .

7. The channel is known at the receiver.

Minimum Mean Square Error (MMSE) equalizer for 2×2 MIMO channel

Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,

.

The received signal on the second receive antenna is,

.

where

, are the received symbol on the first and second antenna respectively,

is the channel from transmit antenna to receive antenna,

is the channel from transmit antenna to receive antenna,

is the channel from transmit antenna to receive antenna,

is the channel from transmit antenna to receive antenna,

, are the transmitted symbols and

is the noise on receive antennas.

We assume that the receiver knows , , and . The receiver also knows and . For convenience, the above equation can be represented in matrix notation as follows:

.

Equivalently,

The Minimum Mean Square Error (MMSE) approach tries to find a coefficient which minimizes the criterion,

The information is very helpful. I have some doubts regarding MMSE.
I am implementing MMSE & ZF with QPSK, 16-QAM and 64 QAM.

1. At higher SNR’s the BER is same for both, is it because the effect high noise will be very negligible at MMSE equalizer?
2. Even in low Snr’s the difference between MMSE and ZF is very less, in the range of 10^-4.
i am using this in combination with OFDM, does this effect the performance?
thanking you in advance

First of all take my gratitude for mailing such a nice demonstration and easy explanation.
I have two queries for you if you have some time please answer.
Why does MMSE provides better performance than Zero-Forcing in terms of system spectral efficiency?
Are the spatial diversity and the antenna diversity same?

Hello Sir,
I ‘m currently trying to implement a MIMO OFDM and MIMO SCFDMA with MMSE equalizer. I ‘ve read your post on OFDM in Rayleight channel where you use convolution to produce multipath. Could you give me an idea on how to do it? Like would you use the model in the MIMO script and within a loop consider subcarrier by subcarrier, then the channel would be convoluted with the transmitted symbol or multiplied? And again how would you combine together your receiver data before the equalization? Can you suggest me anything to read to understand rayleight channel with OFDMA and SCFDMA in MIMO mode and how the subcarrier are treated and equalized at the receiver?

@riccardo:
a) convolution is used to apply the multipath channel on the transmission.
b) if you are doing the modeling in frequency domain, the channel will be multiplied (assuming that the channel taps are within the cyclic prefix)

hello sir,
I am working on enhancement of high speed down link packet access in cellular network by introducing new FUS (far user streaming)scheme. I have seen your huge contribution in the area of communication. I’m facing some problem in simulation with matlab. sir can you please send me some of your matlab code for relay based cellular network
please help me sir…….!devendra.info100@gmail.com

I have been working on frequency domain equalizers and i use single carrier FDE system. However when I compare the performance of Zero Forcing and MMSE equalizers they dont converge at higher SNR. I use a rayliegh random channel with 10 taps. Is this acceptable? I have also sent an email to you.

@Udesh: Please refer to the posts on equalization (zero forcing, mmse) in presence of inter symbol interference athttp://www.dsplog.com/tag/isi/
I also got slightly better performance for MMSE equalizer, but could see that the gap is reducing at higher SNR’s. Try increasing the SNR region to see if it converges at some point.

@Najwen: Hmm… reckon you wish to plot the spectral efficiency in bits/seconds/Hz versus Eb/N0 required to achieve say BER of 10^-5.
One way to go about this is to get the BER vs Eb/N0, dB curve for your modulation scheme with the MMSE receiver. The capacity information is theoretically computed using bit rate and bandwidth. Once you have both the data, we can plug that into the curve.

Dear. Krishna Sankar.
I enjoyed your posts thanks.
I have a doubt, I simulated your simulation by C++, but I found some different results.
With ZF, BER curve fits on your curve. but with MMSE, my BER curve performance degraded by 1 dB. Do you have some idea about this?
Thanks in advance.

@Ari: When you try to reduce the modulation from 4PAM to 2PAM (BPSK), are you seeing differences in the MMSE behaviour vs ZF behaviour. Further, in AWGN the difference might not be present, as there is no interference terms.
Do reply back with your findings.

Thanks for your answer. When i reduce the modulation from 4 PAM to 2 PAM, I seeing differences in the MMSE behaviour vs ZF behaviour like BPSK. Can you explain why in 4 PAM, the difference might not be present ?

Good work !!! The only confusing thing for me in your codes is, why in the step of “Channel and noise Noise addition”, the AWGN is in the format of 10^(-Eb_N0_dB(ii)/20)*n; while in the calculation of a,b,c,d terms, the AWGN is used as 10^(-Eb_N0_dB(ii)/10). Thanks! Jason

Hello
I’d like to know something about MMSE. In the program, the MIMO system has MMSE at the receiving end. The coefficiets of MMSE is derived by the above formula. May I ask that ‘is the formula above for IIR MMSE’. If we use FIR MMSE, what should we modify. What is the difference between FIR and IIR MMSE?
Thank you very much

I am currently working on a project involving MMSE equalisation and have found your code useful. However, I am confused as to why you do the inverse of the matrix manually, would it not be easier to use simple Matlab commands to find the W matrix?

hi all
I have started MSc project in Relaying MIMO, and I have done my first code for a BPSK Signal in rayleigh fading but when try to apply it for MIMO it gives me error in matrix dimension, please see the following code and try to help me if u can, thanxxx.

I want to implement soft input soft output MMSE equalizer for one of my projects. as inputs to the equalizer ,received symbol sequence and LLR value obtained from turbo decoding are fed. from the LLR value fed to the equalizer can generate mean and variance for symbol set(constellation). then using that mean and variance i want to find mean and variance for each symbol. (for simplicity consider BPSK and gaussian distribution of each symbol.). do u hv ay idea about how to do that. i think i hv clariffied my problem enough. Please be kind enough to reply me as am stuckd with this.
further details : Minimum mean squared error equalization using A Priori Information (Michael Tuchler , Andrew C. Singer)

your site has been very useful to study concepts regarding ofdm and mimo. i am currently doing project related to ofdm-mimo. Can you give me some idea.. regarding how to proceed for coding 3×3 mimo or higher….
Thanks in advance..

For the MIMO case with MMSE detection, is the received Eb/No calibrated per transmitted stream? For example, when you go from the 1×2 case to the 2×2 case, does the noise variance stay the same, and you just add a second stream to a second antenna, of power equal to that of the original stream ? Thanks.

hello krishna sankar
do you have matlab code about soft quantization with known noise variance
(chi-square) in ofdm mobile radio system with bpsk modulation ?or any refferances or any site discuss about this?

Hi Krishna, I wrote a matlab program to estimate x through y (y=hx+n) where n is au gaussian noise. I used MMSE to find x through the formula xHat=h’*inv(h*h’+10^(-SNR/10)*I)*y. When i wanted to compare the Theoretical SNR at the receiver (||h||^2 E(x^2)/N0) to the SNR calculated at the output of the MMSE receiver ((1/MMSE)-1), i found the SNR at the output of MMSE is much more higher than the theoretical one. Do you have any explanation on this. Does it mean that the MMSE improve the SNR ? Thank you.

thanks for your your graet work
please I study for my master in frequency synchronization in mimo ofdm system but i have problem with the matlab code to simulate to find out if estimation of the CFO on one path is affected by the CFO values of the adjacent paths and examine the estimator accuracy in term of its mean and variance

If all the chains have a common RF clock, then all the chains will have similar CFO and the estimate from all the chains can be combined to improve the accuracy of the CFO estimation.
If the chains have independent RF clock, then we need to estimate CFO on each chain independently.

thanks for your your graet work
please I study for my master in frequency synchronization in mimo ofdm system but i have problem with the matlab code to simulate to find out if estimation of the CFO on one path is affected by the CFO values of the adjacent paths and examine the estimator accuracy in term of its mean and variance

Signal s—a BPSK signal that takes the values of ±1 with equal probability—passes
through channel c which has the transfer function 1 + 0.5z -1. This means that the output
of the channel at instant ݅ is equal to s(i)+0.5s(i-1). Zero-mean white Gaussian noise
v(i) with variance sigma 2 is added to the channel’s output so that
Y(i)=s(i)+0.5s(i-1)+v(i)
The sequence ݏሺ݅ሻ is white and is independent of noise v.

A three-tap linear minimum mean square error (LMMSE) equalizer is used to estimate
S(i-) using the three samples y(i), y(i-1), y(i-2) . We will solve for the two
cases of  = 0 and  = 1. The LMMSE equalizer forms a linear combination of
y(i), y(i-1), y(i-2) to produce the scalar estimate s^(i- ) such that ሾ[s(i-
) –- s(i- )]2 is minimized.

Find the LMMSE equalizer’s coefficients for the two cases  = 0 and  =1. Note
that the coefficients are a function of sigma 2. Plot the mean square error for both cases of 
on the same graph using the signal-to-noise ratio (SNR) at the input of the equalizer as
the horizontal axis. SNR in this problem is defined as the ratio of signal power to sigma 2at
the input of the equalizer. (Hint: you need to obtain the signal power at the output of the
channel to obtain the power of the signal to which the noise is added.) Plot over the SNR
range from 0 dB to 12 dB.

Hello. Thanks for your posts.
I have a question.
Your simulation result shows that the MMSE equalizer has 3dB improvement than the ZF equalizer.
But, the reference book – [DIG-COMM-BARRY-LEE-MESSERSCHMITT] – shows MMSE detector outperforms ZF detector by 1.8dB from “10.3.9. Performance Comparison”.
Which one is correct?

When I simulated 2by2 MIMO system, the 4QAM-ZF system has same BER of BPSK-ZF system.
(Rayleigh ch./AWGN noise/BER vs SNR per bit per antenna(Eb/No)/4QAM=1/sqrt(2)*{+-1+-j})
But, there is a little difference(almost 1dB) between 4QAM-MMSE and BPSK-MMSE system.(4QAM is worse.) This shows that the MMSE outperforms ZF by 2dB at 4QAM system when the MMSE outperforms ZF by 3dB at BPSK system.
So, I am very confusing.

@JH: The simulations which I did where comparing BPSK ZF with BPSK MMSE (and in the book, comparison is between QPSK-ZF with QPSK-MMSE). I am not sure that’s the reason for the difference in the performance.

seems that we get the best performance (lower BER @ same SNR) without the 2, even if the difference is very small. Anyway this doesn’t mean that the “2″ shouldn’t be there, it’s just a consideration.

But I have another question:

we know that the MMSE should come to have the same performance of the ZF for high SNRs, because the noise term become less relevant.

So, why don’t the two curves (ZF_BER and MMSE_BER) merge?

Well, actually I do have an answer for this, but my supervisor was a little reluctant to accept it:

if you plot the two curves in linear scale they do merge… but in the log scale they run as parallels… I think that’s because the reason why they merge is the SNR ( in DB!!) linearly raising… but a a linear raising of a dB value means a line in log-scale, so there won’t be a slope changing to see the curves merge in log-scale..

Also even if the receiver is different (ZF or MMSE) the system still has some properties that stay the same in the two case: I refer two the slope of the BER curve which for a Ntx=Nrx system is one decade down in 10dB (no diversity, or diversity order = 1). This must be true for both the receivers… so if they have the same slope they won’t merge in log scale.

Let me know what do you think of what I’ve said and if you have some other explanation.

@Franseco: My replies
1/ I rechecked the equations for MMSE. The noise term is E{n*n^H}. The variance of real and imaginary arm of noise is 0.5*10^(-Eb_N0_dB/10). When we compute the noise power, we have to add the variances of real and imaginary term and the total variance is 10^(-Eb_N0_dB/10). Agree?
2/ Well, I also have difficulty accepting the linear vs dB hypothesis. Did you try running the curve for very high values of Eb_N0_dB? Lets say till 100dB?

Hi,
Its a great post indeed.
I am having a very basic doubt. I am just stating the flow for my understanding:
1) Signal ‘X’ is transmitted with pilots.
2) Received signal Y = H.X + N
3) At receiver channel estimation is done i.e ‘H’ is calculated with the help ref. signals/pilots, received).
4) Then ‘W’ is calculated.
5) Now I need to retrieve the original signal (X). Can you please tell me how to do that in any case, zero forcing or MMSE. Is it just linear division or something else is involved. It will be helpful if you may tell the equation as well.

Hi Krishna Pillai
With your model the dimension of the noise correlation matrix shall be Nrx xNrx. But the dimension of the noise correlation matrix in the equalizer is Ntx x Ntx. What is the meaning of this and how can you get it in case of Nrx>Ntx?

@Kartik: Extending the Matlab code to an nRx x nTx case involves modification to the equalizer. In the current code, we have a 2×2 matrix inversion. If we go for higher dimension matrices, we need to change the logic for matrix inversion accordingly. Alternatively, one can use the pinv() function – but then, we loose some of the vectorizing advantages (which results in faster execution) which we now have in the code.

@Venki: I am just guessing, if we know that
y = x + n, where
y is the received pilots,
x is the transmitted reference pilots,
n is the noise.
Then the variance of n can be estimated by finding E { (y-x)^2 }, where E{} is the expectation operator.
Agree?

If i have Reference symbols in frequency domain then can i add N0=(y-x)^2 this directly for Noise Variance Calculation……..bcz i thought noise addition should be in Time Domain……….so do i need to perform IFFT(N0) for Time domain conversion?????????????

@Venki: You can do an ifft(N0) to find the noise in frequency domain. But, the variance of the noise term N0 does not change irrespective of whether we do ifft() or not. Hence doing ifft() is not needed. Agree?

a) Wont MRC have noise enhancement problem? (Because I still dont see clear difference between ZF and MRC)
b) Even if it is SIMO, in real system you will always have interference. So, my answer would be MMSE (with IRC option, where we dont estimatejust noise variance but rather noise+interferece covariance matrix).

@WirelessNewbie: Sorry, the page 141 is not available from the link you provided. Let me try to get the book from the library, and I will respond.
One query: Is the claim that “ZF is better than MMSE for SISO” for a flat fading channel?

Thanks for the previous reply
In the sample code given, the noise variance is n, but it is not used in the receiver, instead of that 10^(-Eb_N0_dB(ii)/10) is used.
Is it because the n is not known to receiver? So we have to measure the SNR in the receiver in practical case?

@WirelessNewbie: In the simulation code, n is the noise voltage signal. For MMSE equalizer, we do not need the noise voltage, rather we only need the variance of the noise. Hence the term 10^(-Eb_N0_dB(ii)/10) is used. Agree?

Yes, a practical MMSE implementation needs to know the measure the SNR at the receiver.

Hi Krishna,
I understand that MMSE is minimizing the equationhttp://www.dsplog.com/cgi-bin/mimetex.cgi?E\left\{%20\mathbf{\left[Wy-x\right]\left[Wy-x\right]}^H\right\}
where can I find, what’s the reason for this? why we choose that equation. (I don’t even know such basics). can you suggest a book.

and in the above comment Parejas is asking you to reuse your post in his italian site(If my translation is correct). You gave him permission ?

@maya: Well, I diversity in the general sense means – the using the extra information which is available and/or transmitted to improve the reliability of the communication link. In general, if we have multiple antennas at the receiver, we have to think of intelligent ways to do receive diversity. Similarly, if we have multiple transmit antennas, we have to figure out ways of processing of the data such that the reliability of the link is increased.

Beamforming is one way of doing transmit diversity, where the knowledge of the channel is used to process the information at the transmitter.

hey sir,can u plzz telme dat how we can find theoretical ber for mimo systems???
and also can u guide me dat can we send a copy of data from 2 transmitters at da same tym instead of selecting a pair of data n den sending it,,is it a right approach???

@maya: Well, from the simulations which I did a 2 transmit 2 receive MIMO V-BLAST system with Zero Forcing Equalizer in flat fading rayleigh channel, performed about identically with a 1 transmit 1 receive system.

Well, if we send the same information from two transmit antennas, I believe there is not much performance gain in flat fading rayleigh channel. I have written a brief post on transmit beamforming, which briefly touches upon this topichttp://www.dsplog.com/2009/04/13/transmit-beamforming/

@safwan: Well, I do not quite understand the need for having an MMSE equalizer for a 2 transmit 1 receive system. If there is no coding at the transmitter, then 2 transmit 1 receive system performs as if its a 1 transmit 1 receive system. Agree?

hCof(1,1,:) = sum(h(2,:,:).*conj(h(:,2,:)),1);
hCof(2,2,:) = sum(h(1,:,:).*conj(h(:,1,:)),1);
hCof(2,1,:) = -sum(h(2,:,:).*conj(h(:,1,:)),1);
hCof(1,2,:) = -sum(h(1,:,:).*conj(h(:,2,:)),1);
when I apply these changes I do not find the same results.

@Wassim: Note that am computing the cofactor of the matrix (H^H*H). Inverse of a [2 x 2] matrix
[a b; c d] = 1/(ad-bc)[d -b;-c a]
The code which you have pasted does not include the matrix rearrangement to compute the cofactor. Agree?

Can you just give me your email address, so that I can mail you my queries. I am working on MIMO OFDM equalization techniques and have a series of questions that I have doubts in. Will you please kindly send your address at cvvarun_raj@yahoo.co.in?

i’m see your script of MIMO equalizer MMSE for 2×2 and my doubt is, why don’t you multiply for 2 the noise? because you consider Es = 1 (ok) but there is 2 antennas. It would want so (10^(-Eb_N0_dB(ii)/20)*2), not?

Dear sir,
I have utilized the MMSE equalizer defined by equation 1./(h.*conj(h)+ 10^(-Eb_N0_dB/10)).*conj(h) for frequency domain equalization of a single carrier modulation system. Now i have to give the reference of this equation in my research work. So kindly tell me the book or any paper which includes so that i could refer this. Plz also provide a soft copy (pdf, web link) if available.

I am working on developing IQ Imabalance compensation scheme in MIMO-OFDM systems. I have done IQ Compensation in OFDM systems but when I am combining MIMO with OFDM, I am not getting proper BER curve and the results r too bad, I have used rayleigh channel and added AWGN noise to the signal transmitted and assumed that the channel is flat, known and the path gains doesnt change for two OFDM symbol durations.I have used Alamouti STC for 2X2 MIMO channel. Please help me getting a solution and send me some related MATLAB codes

@NAVAL: It’s bit hard to say, why the code which is working for 1×1 OFDM is not working for 2×2 OFDM. I think to nail the problem, you should remove awgn, channel etc and make sure that your algorithm is working in ideal conditions. Btw, it sounds like you are trying to do receive I/Q imbalance estimation/compensation? Why do you need the ‘channel is constant for two symbol duration’ assumption.

I agree that it should be slightly better than 1×1 case. What I meant was the comparison between bpsk case and 4 qam case. for zero forcing and 2×2 mimo system both bpsk and 4qam achieve the same performance, but for the mmse case 4qam is slightly worse than bpsk, i suppose it comes from the fact that 4qam is more affected by the noise. Am I right?

Perhaps the title could be spesify because there are several MIMO mode (STBC/STC, BLAST-family,SM, STTC).
If I’m not mistaken, MIMO that your refer in this article is SM Mode (usually using V-Blast).
So, I think the title sould be “Spatial Mutliplex MIMO…” or “V-BLAST MIMO…”

Mr. Krishna, I’ve been making correlated and uncorrelated MIMO channel model based on WiMAX Forum Channel Model Recommendation and ITU Channel Model recommendation. As I know, channel model doesn’t have parameters that can be directly analyst, is our channel model correct or not.

1. Could you give me some advice/ material about steps/process to make MIMO Channel?
2. Could you explain how to analyst correctness of my MIMO channel model?
3. Could you send me example/script of MIMO channel model (correlated/ uncorrelated)? So, I can compare my model with yours.

@andjas: My replies:
1. One great reference is the channel model defined by the High Throughput study group for 802.11n standards development.Tgn Channel Models, Vinko Erceg et al. The document provides a good overview of MIMO channel modeling – includng the effect of antenna correlation (based on antenna spacing). effect of fluorescent lights, doppler for different indoor multipath characteristics.

@jaka: For CDMA, the concept remains the same. However, I would think that to simulate the CDMA MMSE detector case, the Matlab script in this post need to be modified to include:
(a) spreading and depreading,
(b) make flat fading rayleigh channel to a frequency selective channel
(c) remove mimo and make it a single spatial stream.

hi? sir,
when i simulated MMSE 16QAM 2×2 sysem
it’s performance almost same as zero-forcing
is it a right? i used below equation
inv(H^H*H+10^(-SNR/10)*Im)*H^H*R;
(Im:identity matrix, H:channel, R:received
signal inv:inverse)
is it depend on noise variance?
please answer me thanks

@dokich: From the analysis done on BPSK modulation, the MMSE equalier should provide slightly better performance that ZF equalizer (for any value of noise variance). So, there might be some bug in your simulation setup.
Try forcing the noise variance term in the MMSE equalizer to be zero. Then you should get the identical performance to ZF equalizer. Once you obtain that, you may try with different values of noise variance.

hello krishna
i want to simulate a MC-CDMA system that in it calculate PAPR and detect diffrent users with BER calculating and can not do this completly. for this i ask you that if have no problem for you help and give me a matlab file about this problem
thanks a lot for your attention

With a simple Matlab/Octave script may I try to show that ZF and MMSE equalizer gives the same BER. Infact, am not sure whether we can call it as ZF/MMSE equalization – as there is no interference terms.

Thanks for the very interesting blog ..
I am wondering whether it makes any sense to apply MMSE and ML equalization on SISO communication systems in order to get over the ZF equalization major problem of noise enhancement (when channel is in deep fade).
In such a case, how is the theoretical BER performance of BPSK (for instance) over flat rayleigh (given by proakis, Digital communications) affected ? can u reccommend any papers that calculate the theoretical BER performance of ML and MMSE equalization over flat fading .. Most papers i got assume ZF equalization when it comes to SISO systems !

in addition to my first request for investigating the MIMO-OFDM system one question is that how can we maintain the channel coefficients equal for two consecutive symbol(for satisfying the Alamouti decoding scheme) while we are modulating symbols on for example 64 subcarriers of one antenna and the channel has 3-taps?

Thanks Krishna alots! Try to research MIMO, I also have a project with MIMO system: Using Genetic Alogrithms as a tool to handle large numbers of users. I hope we can discuss this project if you don’t mind.
Regards!

Hi & Thanks for your good posts
will you combine the concepts that you have investigated so far step by step and show how is the performance of
MIMO-OFDM systems with alamouti coding and different equalizers such as MMSE in multi-tap Rayligh channel as your conclusion and complex of all of these concepts.

@aydar: My replies:
a) I do not have a precise understanding of how MRC works in the case of 3 receive 2 transmit case. I am sure that having the ‘extra’ antenna at the receiver improves the diversity gain. However, am not sure what we are doing is MRC or not
b) Why do you say that there is interference in SIMO systems?

“…The CAZAC sequence is a sequence that has constant amplitude in both regions of time and frequency and it has always Zero Auto-Correlation for time shift that is other than the cyclic self-correlation value is 0. As the CAZAC sequence has Constant amplitude in a time region, it can keep PAPR (Peak to Average Power Ratio) low. As the CAZAC sequence also has Constant amplitude in a frequency region, it is a sequence suitable for propagation path estimation in the frequency region. Here, a small PAPR means that it can keep the power consumption low. This feature is preferred in the mobile communication. …”

Helleo sir
I am doing project on Bit Error performance of MIMO system. Pls send me comparative BER analysis of MIMO system for different modulation technique such as BPSK QPSK and QAM. This information is very useful for me. Pls send it for me.